| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Modulus and argument with equations |
| Difficulty | Standard +0.3 This is a straightforward Further Maths FP1 question requiring algebraic manipulation of complex numbers (cross-multiplying and equating real/imaginary parts), then using the argument condition to find λ via a simple tangent equation, and finally computing |z|². All techniques are standard for FP1 with clear signposting and no novel insight required. Slightly above average difficulty due to being Further Maths content, but routine within that context. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
Given that $\frac{z + 2i}{z - \lambda i} = i$, where $\lambda$ is a positive, real constant,
\begin{enumerate}[label=(\alph*)]
\item show that $z = \left( \frac{\lambda}{2} + 1 \right) + i \left( \frac{\lambda}{2} - 1 \right)$. [5]
\end{enumerate}
Given also that $\arg z = \arctan \frac{1}{3}$, calculate
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item the value of $\lambda$, [3]
\item the value of $|z|^2$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 Q28 [10]}}